Initial classification of global behavior around multi-solitons for the nonlinear Klein-Gordon equation

Kenji Nakanishi (Kyoto U)

May 15. 2024, 10:30 — 11:30

This is joint work with Gong Chen (GATech). We study the nonlinear Klein-Gordon equation with the focusing cubic power in three space dimensions. Our goal is to classify the initial data for the global behavior in an open neighborhood of multi-solitons generated from the ground state. The main difficulty is to trace the unstable mode of each soliton, whose exponential growth rate depends on its speed. An ODE model suggests that the most unstable mode can screw up the growth rates and directions of the slower ones, even if their interactions are indirect with exponential decay factors. It may sound like the end of story, as the dispersive estimates do not decay exponentially, but our situation turns out to be better. The key mechanism is growth delay during energy transfer between solitons, and our key tool is an energy estimate with exponential space-time weight for the linearized equation around mutli-solitons.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Recordings:
Recording
Associated Event:
Nonlinear Waves and Relativity (Thematic Programme)
Organizer(s):
Roland Donninger (U of Vienna)
David Fajman (U of Vienna)
Birgit Schörkhuber (U of Innsbruck)